Question

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Explanation:

Step1: Recall Tangent-Chord Angle Theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. For \(\angle VUX\), the intercepted arc is \(\overset{\frown}{UV}\) with \(m\overset{\frown}{UV} = 62^\circ\). So, \(m\angle VUX=\frac{1}{2}m\overset{\frown}{UV}\).

$$m\angle VUX=\frac{1}{2}\times62^\circ = 31^\circ$$

Step2: Recall Diameter and Arc Relationship

Since \(UW\) is a diameter, \(m\overset{\frown}{UVW}=180^\circ\). We know \(m\overset{\frown}{UV} = 62^\circ\), so \(m\overset{\frown}{VW}=180^\circ - 62^\circ=118^\circ\)? Wait, no, for \(\angle VUW\), it's an inscribed angle? Wait, no, \(\angle VUW\) is an inscribed angle? Wait, no, \(UW\) is diameter, \(UV\) is chord. The measure of \(\angle VUW\): wait, no, actually, \(\angle VUX\) is tangent-chord, and \(\angle VUW\): since \(UX\) is tangent, \(UX\perp UW\) (tangent to a circle is perpendicular to the radius at the point of tangency, and \(UW\) is diameter, so radius at \(U\) is along \(UW\), so \(UX\perp UW\), so \(\angle XUW = 90^\circ\). Then \(\angle VUW=90^\circ - \angle VUX\). Wait, from Step1, \(\angle VUX = 31^\circ\), so \(\angle VUW = 90^\circ - 31^\circ = 59^\circ\)? Wait, no, another way: the measure of an inscribed angle is half the measure of its intercepted arc. \(\angle VUW\) intercepts \(\overset{\frown}{VW}\), but wait, \(UW\) is diameter, so \(\overset{\frown}{UV}+\overset{\frown}{VW}=180^\circ\), so \(\overset{\frown}{VW}=180 - 62 = 118^\circ\)? No, that's not right. Wait, no, \(\angle VUW\): actually, the angle between chord \(UV\) and diameter \(UW\). Wait, the tangent-chord angle is half the intercepted arc, so \(\angle VUX = 31^\circ\), and since \(UX\perp UW\) (tangent perpendicular to diameter), \(\angle XUW = 90^\circ\), so \(\angle VUW = 90^\circ - \angle VUX = 90 - 31 = 59^\circ\). Alternatively, the inscribed angle over arc \(VW\): wait, no, arc \(UV\) is \(62^\circ\), so the central angle for arc \(UV\) is \(62^\circ\), and \(\angle VUW\): let's think again. The tangent \(UX\) and chord \(UV\) form \(\angle VUX = 31^\circ\) (half of \(62^\circ\)). Since \(UX\) is tangent, \(UX\perp UW\) (because \(UW\) is diameter, so radius at \(U\) is \(UW\) direction, tangent is perpendicular to radius), so \(\angle XUW = 90^\circ\). Therefore, \(\angle VUW = \angle XUW - \angle VUX = 90^\circ - 31^\circ = 59^\circ\). Wait, but also, the inscribed angle over arc \(VW\): arc \(VW\) is \(180 - 62 = 118^\circ\), so inscribed angle would be half, \(59^\circ\), which matches. So that's correct.

Answer:

s:
(a) \(31\)
(b) \(59\)